An equivalent criteria for irrationality of ζ(5)

2023/09/09 by Suman, Shekhar
#11J72 #11M06 #FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.2312.00298

Abstract

Defining a Beukers [1] like integral for ζ(5) as I_n:=∫_(0,1)⁵((1-x₃)ⁿ(1-x₄)ⁿ P_n(x₁)P_n(x₂))/(1-(1-x₁x₂x₃x₄)x₅) dx₁dx₂dx₃dx₄dx₅ we prove that for each n∈ℕ I_n= (p_nζ(5)+q_nζ(4)+r_nζ(3)+s_n)/(d_n⁵) where p_n,q_n,r_n,s_n are integers and d_n=lcm(1,2,...,n). We prove that the following are equivalent: 1. q_nζ(4)+r_nζ(3)-d_n⁵ I_n∉ℤ for each natural number n. 2. q_nζ(4)+r_nζ(3)-d_n⁵ I_n∉ℤ for infinitely many natural number n. 3. ζ(5) is irrational.

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